Quantum phase transition of Ising-coupled Kondo impurities

M. Garst,1 S. Kehrein,2 T. Pruschke,2 A. Rosch,1 and M. Vojta11Institut für Theorie der Kondensierten Materie, Universität Karlsruhe, Postfach 6980, 76128 Karlsruhe, Germany

2Theoretische Physik III, Elektronische Korrelationen und Magnetismus, Universität Augsburg, 86135 Augsburg, Germany(Received 15 October 2003; revised manuscript received 11 February 2004; published 18 June 2004)

We investigate a model of two Kondo impurities coupled via an Ising interaction. Exploiting the mapping toa generalized single-impurity Anderson model, we establish that the model has a singlet and a(pseudospin)doublet phase separated by a Kosterlitz-Thouless quantum phase transition. Based on a strong-coupling analy-sis and renormalization-group arguments, we show that at this transition the conductanceG through the systemeither displays a zero-bias anomaly,G,uVu−2sÎ2−1d, or takes a universal value,G=se2/p"dcos2sp /2Î2d, de-pending on the experimental setup. Close to the Toulouse point of the individual Kondo impurities, thestrong-coupling analysis allows us to obtain the location of the phase boundary analytically. For general modelparameters, we determine the phase diagram and investigate the thermodynamics using numericalrenormalization-group calculations. In the singlet phase close to the quantum phase transition, the entropy isquenched in two steps: first the two Ising-coupled spins form a magnetic minidomain which is, in a secondstep, screened by a Kondoesque collective resonance in an effectivesolitonicFermi sea. In addition, we presenta flow-equation analysis which provides a different mapping of the two-impurity model to a generalizedsingle-impurity Anderson model in terms of fully renormalized couplings, which is applicable for the wholerange of model parameters.

DOI: 10.1103/PhysRevB.69.214413 PACS number(s): 75.20.Hr, 73.21.La, 71.10.Hf

I. INTRODUCTION

Kondo physics plays a fundamental role for the low-temperature behavior of a large variety of physical systemssuch as magnetic impurities in metals, heavy fermion sys-tems, glasses, quantum dots, etc. Its key feature is thequenching of the impurity entropy through nonperturbativescreening by many-particle excitations in the associatedquantum bath. For magnetic impurities in metals thisamounts to the formation of the Kondo singlet between thelocalized spin and electron-hole excitations in the Fermi sea.1

Most of the aspects of single-impurity Kondo physics arenow well understood after theoretical tools have been devel-oped that can deal with its intrinsic strong-couplingnature.1–3 However, in many physical systems the interactionof different impurities, i.e., multi-impurity Kondo physics, isimportant. For example, in heavy fermion systems theRuderman-Kittel-Kasuya-Yosida(RKKY ) interaction be-tween different impurity spins leads to competition betweenlocal Kondo physics and long-range magnetic order that de-termines their phase diagram.4 More recently, related ques-tions about coupled two-level systems have gained much in-terest in quantum computation, where decoherence due tounwanted couplings among qubits and between qubits andenvironment should be avoided; on the other hand the inten-tional coupling of qubits is the key step to perform quantumlogic operations.

In the present paper we investigate the case of two spin-1/2 Kondo impuritiesS1,S2 coupled via an Ising coupling,

H12Ising = KzS1

zS2z. s1d

The Kondo coupling of each impurity to its bath is given by

HjK = 2J'sSj

xs0,jx + Sj

ys0,jy d + 2JzSj

zs0,jz , s2d

where j =1,2 labels the impurity, ands0,j is the bath spinoperator at the respective impurity site. Furthermore the twobaths are disconnected.

Two impurities coupled both to baths and among eachother present the simplest realization of the so-called clusterKondo effect, which has been discussed, e.g., in context ofdisordered Kondo-lattice compounds.5 Furthermore, the dy-namics of magnetic droplets or domains, formed in disor-dered itinerant systems near a magnetic quantum phasetransition,6 also leads to models of coupled impurities suchas the one considered here. Therefore, we will also refer tothe two coupled impurities as magnetic “minidomain.”

In the context of Kondo impurities, an Ising-like coupling(1) can be thought of as an effective impurity interaction forheavy fermion systems with an easy axis. Also, Ising cou-pling appears naturally in quantum dots that are coupled viatheir mutual capacitance7 here the two-level systems arepseudospins representing the number of electrons on thedots, and therefore SU(2) symmetry is broken from theoutset.8–10 Equivalently, one can think of two two-level sys-tems with transversal coupling, with the experimental real-ization of coupled flux qubits.11 We will discuss differentformulations and applications of our model in the body ofthe paper.

Coupled impurities or two-level systems have been inves-tigated in a number of papers,12–19 where most attention hasbeen focussed on the case of SU(2)-symmetric direct ex-change coupling between the impurity spins,KS1·S2. Here,two different regimes are possible as a function of the inter-impurity exchangeK: for large antiferromagneticK the im-purities combine to a singlet, and the interaction with theconduction band is weak, whereas for ferromagneticK the

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impurity spins add up and are Kondo screened by conductionelectrons in the low-temperature limit. Notably, there isnoquantum phase transition asK is varied in the generic situa-tion without particle-hole symmetry(whereas one finds anunstable non-Fermi-liquid fixed point in the particle-holesymmetric case).13,14,16

As has been pointed out by Andreiet al.,20 the case ofIsing coupling is different and particularly interesting, be-cause for largeuKzu the two Ising-coupled spins form a mag-netic minidomain which still contains an internal degree offreedom as the ground state ofH12

Ising is doubly degenerate(incontrast to the interimpurity singlet mentioned above). Forthe case of antiferromagneticKz (which we will assume inthe following), the two low-energy states of the impurities(forming a pseudospin) areu↑↓l andu↓↑l. As we will show inthis paper, the fate of this pseudospin degree of freedomdepends on the strength and asymmetry of the Kondo cou-pling J between the spins and the bath electrons.

A. Summary of results

Here, we will summarize our main results which are de-tailed in the body of the paper, and schematically representedin the phase diagram of Fig. 1. A brief summary of the meth-ods used to obtain these results is given below in Sec. I B.

The model of two Ising-coupled impurities, connected totwo separatefermionic reservoirs(realized, e.g., by attach-ing two separate leads to two quantum dots, Fig. 2), has twoground-state phases associated to either a screened or an un-screened pseudospin. For small Kondo couplingsJ', Jz andlarge Kz, tunneling between the two pseudospin configura-tions, u↑↓l and u↓↑l, is suppressed at low energies, i.e., theminidomain is “frozen” asT→0, and the ground-state en-tropy is S0= ln2. In contrast, for smallKz the two impuritiesare individually Kondo screened, resulting in a Fermi-liquidphase with vanishing residual entropy. This implies the exis-tence of a quantum phase transition forKz,TK

s1d, whereTKs1d

is the single-impurity Kondo temperature. For isotropicKondo coupling, i.e., smallJz, this has been previouslypointed out by Andreiet al. 20

What is the nature of this phase transition? Does it occurby breaking up the Ising-coupled minidomain or rather bystrong fluctuations of the preformed pseudospin? What arethe universal properties of this transition? The key observa-tion, which helps to answer these questions, is that the sys-tem can also be tuned towards the quantum phase transitionby increasing the Ising componentJz of the coupling of thespins to the environment in a regime whereKz@TK

s1d (seeFig. 1). For Kz@TK

s1d the minidomain is stable. However,upon increasingJz a many-particle effect(related to forma-tion of a Mahan exciton21) enhances the tunneling betweenthe two configurations,u↑↓l and u↓↑l, of the minidomain. IfJz exceeds a critical value,Jz

cr, this tunneling can quench thepseudospin even for infinitesimalTK

s1d (equivalent to infini-tesimal transverse Kondo couplingJ').

The resulting phase of the “fluctuating minidomain” isactually a Fermi liquid with vanishing residual entropy. Notethat the finite-temperature properties in this regime are rather

different from that of the Fermi liquid which is obtained forTK

s1d@Kz. For largeJz and smallJ', the high-temperature ln4impurity entropy is quenched in two stages: first, at the scaleT0

screening.(Note that this type of two-stage screening is com-pletely different from the one occurring for two conventionalKondo screening channels with different strengths.12) In con-trast, forTK

s1d@Kz the entropy of the two spins is quenchedsimultaneously in a single step at the scaleTK

s1d. Nevertheless,the two Fermi-liquid regimes are adiabatically connected bya smooth crossover, which we will show to be identical tothe well-known crossover in the single-impurity Andersonmodel from the mixed valence into the Kondo regime.

The quantum phase transition atJz=Jzcr turns out to be in

the Kosterlitz-Thouless universality class. Assuming conti-nuity along the phase boundary(which we verified numeri-cally), this is true for the transition at arbitraryJzøJz

cr. Fur-thermore, universality immediately implies that thefluctuating minidomain regime with its characteristic two-stage quenching of the entropy also exists for smallJz closeto the quantum phase transition(see Fig. 1).

Physical observables, such as the conductance in a quan-tum dot setup, showuniversalbehavior in the vicinity of thephase boundary. Therefore, we can calculate them close toJz

cr (see Fig. 1), where the phase transition takes place in aregime of largeKz being accessible to a strong-couplinganalysis combined with renormalization-group arguments.Depending on the experimental setup, we find, e.g., a con-ductance anomaly characterized by the exponent −2sÎ2−1dor a universal conductancee2 cos2fp /2Î2g / s"pd at thephase transition. We emphasize again that these results arevalid close to the quantum phase transition even for smallJzwhere a strong-coupling analysis is not possible.

B. Methods and outline

To obtain the physical picture and the results describedabove, we use a combination of six different and partlycomplementary methods.

In Sec. II we(i) map our model of Ising-coupled spins toa generalized Anderson model by bosonization and refermi-onization techniques. This mapping is used to obtain thequalitative structure of the phase diagram and analytic resultsfor the phase boundary at largeJz. Furthermore, for the gen-eralized Anderson model it is much easier to implement(ii )numerical renormalization group(NRG), which is presentedin Sec. IV. With the help of NRG it is possible to determinenumerically the phase boundaries in regimes not accessibleto analytic methods. Furthermore, NRG is essential to estab-lish that the phase transition at smallJz is continuously con-nected to the one at largeJz.

Making use of this adiabatic continuity is the main idea ofthis paper to obtain analytic results for the quantum phasetransition. By increasingJz we can tune the transition from aregime withKz,TK

s1d to a regime withKz@TKs1d, where we

can employ(iii ) a strong-coupling expansion(Sec. III). Thestrong-coupling result is analyzed using(iv) perturbativerenormalization group, or more precisely power counting,taking into account the anomalous dimensions created froman orthogonality catastrophe. Using these methods the phasediagram for largeKz@TK

s1d and the precise position of thecritical point forKz→` can be obtained. In addition, we can

determine the relevant phase shifts and scaling dimensions ofleading relevant and irrelevant operators(Sec. VI). This al-lows us to analytically calculate the conductance and zero-bias anomalies close to the quantum phase transition forar-bitrary values ofTK

s1d /Kz, see Sec. VII. To analytically obtainthe precise shape of the phase diagram for largeJz, we de-velop in Appendix A(v) a generalization of the Schrieffer-Wolff transformation to take into account the short-rangedensity interaction of our generalized Anderson model, lead-ing to associated power-law singularities.

In Sec. V and Appendix B we re-derive some of the aboveresults independently by using(vi) flow equations. The ad-vantage of this method is that it has a broad range of appli-cability and gives a more natural description in terms ofrenormalized quantities. The flow-equation mapping nicelyestablishes the equivalence of the different Fermi-liquid re-gimes of the model. It also allows us to derive the full phasediagram analytically for general values of the couplingJzthat are not accessible with the strong-coupling expansion.

Transport quantities and corresponding possible experi-mental realizations are discussed in Sec. VII. The mostpromising way to implement Ising-coupled(pseudo)spinvariables is the use of charge degrees of freedom of twoquantum dots, following Matveev’s proposal,8,9 and employa capacitive coupling which takes the role ofKz, see Sec.VII A.

Most of our methods are based on the mapping of theoriginal two-impurity model to a generalized single-impurityAnderson model, except for the strong-coupling analysis inSec. III A which is applied to the original model(and thusdirectly establishes the existence of a phase transition). Aswe will show, the employed mapping provides a particularlyclear picture of the underlying physics, e.g., it establishes theuniversality class of the transition, and allows us to makefurther progress using flow equations. Thus, the mappingturns out to be extremely helpful for obtaining the completepicture presented below.

II. MODEL: VARIATIONS AND TRANSFORMATIONS

In this section we discuss the various formulations of themodel under consideration, together with the mapping be-tween them, which is based on the well-known relation be-tween the spin-boson model and the anisotropic Kondomodel.3,22,23

Throughout this paper, we will consider the so-calledscaling limit where bothKz and the single-impurity KondotemperatureTK

s1d are much smaller than the high-energy cut-off D of the theory,TK

s1d ,Kz!D. KeepingJzù0 andD fixed,this implies J'→0. Only in this scaling limit the modelsdiscussed below can be mapped upon each other. In general,the position of the phase boundary, i.e., the value of thecritical couplingKz

cr depends on microscopic details. In thescaling limit, however,Kz

cr /TKs1d depends only onJz which

parametrizes the renormalization flow in the single-impuritymodel, see Sec. IV C. This universality is represented in thephase diagram in Fig. 1.

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A. Ising-coupled Kondo impurities

We consider the model

HK = H1K + H2

K + H12Ising, s3d

where two spinsS1 and S2 interact by the Ising interaction(1), H12

Ising. Each of the spins couples to a separate fermionicbath, cka j, via an anisotropic Kondo HamiltonianHj

Ks j=1,2d, see Fig. 2,

HjK = H0fCa,jg + o

nab

JnSjnCa j

† s0dsabn Cb js0d, s4d

where a ,b are spin indices andH0fCa,jg=ok,a ekcka j† cka j

with Cs jsxd=ok e−ikxcks j. The exchange coupling is assumedto be the same for both impurities and has an anisotropicform, Jn=sJ' ,J' ,Jzd.

The fermionic baths are assumed to be particle-hole sym-metric with a bandwidthD; for a rectangular band the den-sity of states at the Fermi level is thenrF=1/D. Our conclu-sions are not modified by the presence of a particle-holeasymmetry. A comprehensive discussion of possible modifi-cations of our Hamiltonian(3), e.g., due to tunneling be-tween the fermionic baths, and how they influence our re-sults, will be given in Sec. VI.

A model of form(3) may be approximately realized withreal spins in the presence of a strong Ising anisotropy. Inaddition, it occurs naturally as a model for capacitivelycoupled quantum dots,8–10,20where the local operatorsS1

z andS2

z describe charge states, i.e., pseudospin degrees of free-dom, on the two dots. Concrete application of our results tosuch a situation will be discussed in Sec. VII.

B. Coupled qubits in ohmic baths

An alternative starting point for our model of Ising-coupled impurities can be formulated in terms of two two-level systems(spin-boson models), HSB=H1

SB+H2SB+H12

trans

with

HjSB = H0fbkjg +

D

2s j

x +1

2 ok.0 lks jzsbkj + bkj

† d, s5d

whereH0fbkjg=ok.0 vkbkj† bkj, and a transversal coupling be-

tween them,

H12trans=

Kz4

s1zs2

z. s6d

Herebkj† are the bosonic creation operators for heat bathj . D

is the bare tunneling matrix element between the two levels.The impurity properties are completely parametrized by thespectral functionJsvd;ok lk

2dsv−vkd, which we assume tobe of ohmic form,Jsvd=2ave−v/2vc.

One realization of this model is the interaction of tunnel-ing centers in glasses through higher-order phononexchange.24 In the context of quantum computation thismodel arises in studies of decoherence of coupled supercon-ducting qubits: the transversal couplingKz is generatedthrough a superconducting flux transporter, and the heatbaths describe the environment leading to decoherence.11,25

Here, the assumption of two different baths for the two qu-bits is justified, e.g., if the baths model electromagnetic noisecoming from read-out circuits, which are separate for eachqubit.

C. Bosonization

The equivalence ofHK andHSB can be explicitly shownin the framework of bosonization. Furthermore, we willdemonstrate that both Hamiltonians can be mapped to a gen-eralized Anderson impurity model. We will use this mappingextensively, both to solve the models numerically withinNRG and to identify the position and nature of the quantumphase transition in certain limits analytically.

It is well known22 that both the Kondo model and thespin-boson model are equivalent to a generalized resonant-level model to be defined below. Our model,HK and HSB,however, consists of two coupled Kondo- and spin-bosonHamiltonians. The crucial point is that the assumed couplingH12

Ising and H12trans, respectively, is transformed trivially by

switching between these three representations.We start from the two Kondo Hamiltonians(4) and apply

the bosonization identity

Cs jsxd =1

Î2paFs j e

−ifs jsxd, s7d

wherea is a short-distance cutoff,Fs j is an anticommutingKlein factor shFs j

† ,Fs8 j8j=2d j j 8dss8d, and fs j is the corre-sponding bosonic field withffs jsxd ,]x8fs8 j8sx8dg=2pidsx−x8dd j j 8dss8. Transforming to bosonic charge and spin fields,fs/c,j =s1/Î2dsf↑ j ±f↓ jd, the bosonized version of the KondoHamiltonians(4) reads

HjK = H0ffcjg + H0ffsjg +

JzÎ2p

Sjz]xfsjs0d

+J'

2pase−iÎ2fsjs0dSj

+F↓ j† F↑ j + H.c.d, s8d

whereH0ffg=vFesdx/2pd1/2s]xfd2 assuming a linear dis-persion,ek=vFk. The bosonic charge fieldfcj decouples andis omitted in the following.

Applying a general Emery-Kivelson transformation,23

Ug = expSigoj

Szjfsjs0dD , s9dparametrized byg, the HamiltonianHj

K transforms intoH̃jK

=UgHjKUg

†,

H̃jK = H0ffsjg + S JzÎ2p − gvFDSjz]xfsjs0d

+J'

2pase−isÎ2−gdfsjs0dSj

+F↓ j† F↑ j + H.c.d. s10d

Importantly, the Ising coupling(1) is not affected by thistransformation,H12

Ising=UgH12IsingUg

†.For two special values of the transformation parameterg

the Emery-Kivelson transformation results in particularly in-

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teresting forms of the Hamiltonian. First consider the casewheng=Î2. The exponents in the spin-flip term of Eq.(10)then vanish andH̃j

K can be cast into the form of the spin-boson Hamiltonian(5). We can now easily identify the cou-pling constants,D=J' /pa, lk=sJz/Î2p−Î2vFdÎ2pk/L, vk=vFk and bqj are the Fourier components of]xfsjsxd withfsjsxd=−ok.0 Î2p /kLs−ibkje−ikx+ ibkj† eikxde−ka/2. The lineardispersion and the form of the couplinglq result in a ohmicform of the spectral functionJsvd=2ave−v/2vc with astrength

a = sJzr − 1d2, s11d

wherer is the density of states,r=1/s2pvFd.The single-impurity Kondo temperature has in general a

power-law dependence on the “tunneling rate”J',

TKs1d ~ J'

1/s1−ad sfor J' ! Jzd, s12d

with a introduced above.

D. Generalized Anderson impurity model

Applying the Emery-Kivelson transformation withg=Î2−1 results in exponentials in Eq.(10) having the sameform as in the bosonization identity(7) and can therefore beexpressed as fermionsC j. The refermionized Hamiltoniancan be identified with a generalized resonant-level model26,27

HjRL = H0fC jg + V„dj

†C js0d + H.c.…

+ WSdj†dj − 12D:C j†s0dC js0d:, s13dwhere Sj

z=dj†dj −1/2, V=J' /Î2pa, and W=Î2Jz−sÎ2

−1d /r. C andd are fermionic operators, whereC representssolitonic spin excitations of the original conduction band andd describes the spin degree of freedom of the impurity. Thecoupling W vanishes forJzr=1−1/Î2 (or a=1/2 for thespin-boson model), the so-called Toulouse point of theKondo model;28 in this case Eq.(13) reduces to the conven-tional resonant-level model. Furthermore,W,0 for isotropicsmall Kondo couplings,Jz=J'!D.

In the new variables the Ising interaction takes the formKzsd1

†d1−1/2dsd2†d2−1/2d. If we interpret the bath indexj

=1,2 as apseudospin indexs= ↑ ,↓, we can identify thetotal Hamiltonian (3) with a generalized single-impurityAnderson model,

HA = H0fCsg + Vos

fds†Css0d + H.c.g + Kzn̄d↑n̄d↓

+ Wos

n̄ds:Cs†s0dCss0d:, s14d

with n̄ds=ds†ds−1/2. In this representation, the Ising inter-

action translates to a local Coulomb repulsion, andW corre-sponds to an interaction of the localized levelds with thesurrounding electrons. In the limitKz=0 the Hamiltoniandescribes the extensively studied x-ray threshold problem.21

On the other hand, at the Toulouse point whereW=0, thestandard impurity Anderson Hamiltonian is recovered. For

large Kz the d level is mainly singly occupied; its “spin”scorresponds precisely to the pseudospin degree of freedom ofthe original minidomain. Note that the particle-hole symme-try of the effective Anderson model corresponds to the sym-metry under a rotation byp around thex axis in spin spacefor the original model.

The mapping of the original two-impurity model onto thegeneralized Anderson model(14) is one of the central resultsof our paper, and will be extensively used in the numericalstudy of the phase diagram and the interpretation of the re-sults.

E. Parameter mapping via phase shifts

It is important to note that the precise relation of the threemodelsHSB, HK, andHA depends on the cutoff structure, i.e.,on properties at high energies and short distances. All formu-las quoted above which relate the various coupling constantsare actually only valid within the cutoff scheme underlyingbosonization. However, it is generally believed that all threemodels are equivalent independent of the cutoff structure, aslong as one considers only the universal low-energy proper-ties in a regime whereKz andTK

s1d are much smaller than anyother scale.

We consider now the(nonuniversal) mapping of modelparameters within different cutoff schemes. For small valuesof J' (or D and V) it is possible to calculate the precisemapping by investigating the perturbation theory inJ', D,andV, respectively, using the fact that all three models maponto a Coulomb gas22,23,29—we will not attempt this herebecause it is difficult to do it analytically for an arbitrarycutoff scheme. However, the mapping ofJz, W, anda can beobtained directly by matching the conduction-electron phaseshifts in the limit J' ,V=0, as phase shifts are measurablelow-energy properties.

In the Kondo model(4), we denote the scattering phaseshift for antiparallel conduction electron and impurity spinsby dJz; for parallel spins the phase shift is then −dJz. Analo-gously, the phase shift in the resonant-level model(13) is dWif the d level is unoccupied and −dW if the d level is occu-pied. For a clear distinction, here and in the following wedenote byrF a density of states of a fermionic band withfinite cutoff, whereasr refers to a density of states within thecutoff scheme underlying bosonization. In the latter scheme,the phase shifts defined above are directly proportionalto the coupling constants,dJz=pJzr /2 and dW=pWr /2,where the density of statesr=1/s2pvFd. If one uses insteada model where the high-energy cutoffs arise from a bandstructure, one obtains dJz=arctans−Jz/2dImg00s0d / f1−s−Jz/2dReg00s0dg, whereg00svd=ok 1/sv−ek+ i0+d is thelocal Green’s function of the electrons. In case of particle-hole symmetric bands,g00s0d=−iprF, this relation simplifiesto

dJz = arctanfpJzrF/2g. s15d

Similarly, W in Eq. (14) induces forV=0 a phase shiftdW=arctanfpWrF /2g. Matching the various models by their

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phase shifts, the relationWr=Î2Jzr−sÎ2−1d derived withinbosonization, translates into

dW = arctanfpWrF/2g = dJzÎ2 − p

2sÎ2 − 1d. s16d

Note that this equation is only valid for smallJ' andV.

III. STRONG-COUPLING ANALYSIS

In this section we analyze the behavior of the system forsmall TK

s1d /Kz. After presenting a general argument for theexistence of a phase transition, we discuss the resulting phys-ics in terms of the generalized Anderson model(14). Inter-estingly, twodifferent strong-coupling limits emerge whichwill be described in Secs. III C and III D. As detailed below,both strong-coupling limits display a phase transition of theKosterlitz-Thouless type. Furthermore, the limits will beshown to commute, and the physical regimes are smoothlyconnected.

A. Effective Hamiltonian

To investigate the phase diagram sketched in Fig. 1 weconsider first the limit ofuKzu→`. We can restrict the con-siderations toKz→ +`, as results forKz→−` are similarbecause thez component of the total spin is conserved sepa-rately for the “1” and “2” subsystems.

In the limit Kz→ +` the two-impurity spins form an an-tiferromagnetic minidomain, with configurationsu↑↓l andu↓↑l. No fluctuations can occur forKz=` (or J'=0), there-fore the ground state of the full system is a doublet.

We now set up a perturbation theory in the small param-eter J' /Kz by deriving an effective Hamiltonian in thehu↑↓l , u↓↑lj subspace of the impurities. The lowest processconnecting the two statesu↑↓l, u↓↑l is OsJ'2 /Kzd. Thus theeffective Hamiltonian in the strong Ising-coupling limit reads

HeffK = Heff,0

K + Heffflip , s17d

where Heff,0K is given by H1

K+H2K with the perpendicular

Kondo coupling set to zero,J'=0. Note that the size ofJzcan be arbitrary. The minidomain is flipped by the term

Heffflip =

4J'2

KzsS1

+S2−C↓1

† C↑1C↑2† C↓2 + H.c.d, s18d

with Cai =ok ckai. The zero-temperature stability of the fro-zen minidomain now depends on whether the operatorHeff

flip

is relevant in the renormalization-group sense.SinceHeff

flip is comprised of four electron operators, its bare(tree-level) scaling dimension is negative, dimfHeff

flipgtree=−1.This might suggest that the doublet ground state with re-sidual entropy ln 2 is stable. However, in the present problemHeff

flip acquires an anomalous scaling dimension which modi-fies this conclusion. This can be understood as follows: ForlargeJz a flip of the minidomain suddenly changes the phaseshifts of all electrons in the leads, thus exciting an infinitenumber of particle-hole pairs. This is the well-known or-thogonality catastrophe,30 leading to an anomalous long-time

response of the electrons. In the presence of a sharp Fermiedge this results in a so-called x-ray edge singularity whichis reflected in an anomalous scaling dimension ofHeff

flip (18).In the following we will determine this scaling dimension

using Hopfield’s rule of thumb,31 and identify the criticalJzwhere theHeff

flip becomes relevant, resulting in “quantummelting” of the frozen minidomain.

To adjust the Fermi sea to a new ground state after thedomain has flipped once, a certain amount of chargeDn hasto flow to infinity. Hopfield noticed that collective responseof a Fermi sea depends in the long-time limit only on thisDn: the corresponding correlation function decays ast−sDnd

2.

In our problem, we have to consider four different Fermisurfaces(j =1,2, s= ↑ ,↓) each contributing independently.

A domain flip is induced by the operatorA=S1

+S2−C↓1

† C↑1C↑2† C↓2. According to Hopfield’s rule of

thumb, the correlation function in theabsenceof domainflips is then given by

kA†stdAs0dlHeff,0K , t−a, s19d

where

a = oj=1,2;s=↑,↓

Dnjs2 . s20d

The transferred chargesDnjs are easily obtained from thephase shifts using Friedel’s sum rule. For example, a spin flipinduced byA changes the phase of the down electrons inbath 2 from dJz to −dJz which corresponds according toFriedel’s sum rule to a charge transfer of 2dJz/p. Further-more, the annihilation operatorC↓2 eliminates one chargeand the total charge transfer in this channel is given byDn↓2=2dJz/p−1. Similar arguments giveDn↑2=−Dn↑1=Dn↓1=−Dn↓2. Therefore, the exponenta is given by a=4s2dJz/p−1d

2. This result can also be verified explicitly bybosonization following Schotte and Schotte.32

From Eq. (19), we can directly read off the anomalousscaling dimension of the domain-flip Hamiltonian(18) withrespect to the “frozen-domain” fixed point:

dimfHeffflipg = 1 −

a

2= 1 − 2S2dJz

p− 1D2. s21d

Here, the first term arises from the engineering dimension ofHeff

flip. For small scattering phase shifts dimfHeffflipg is negative,

i.e., the domain-flip Hamiltonian is irrelevant and the doublydegenerate frozen-domain fixed point described byHeff,0

K isstable. Domain flips become relevant fordJz.dT with

dT =p

2S1 − 1Î2D . s22dBeyonddT the fluctuations of the minidomain grow towardslow energies giving rise to a new phase—in this regime thepseudospin describing the minidomain is still well defined,but is ultimately screened at low energies. The special valueof dT is well known as the Toulouse point of the single-impurity Kondo Hamiltonian.33

We have thus established the existence of a phase transi-tion in the limit of J' /Kz!1, which is accessed by varying

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Jz. In the bosonization cutoff scheme, the critical value isrJz

cr=1−1/Î2 corresponding to the Toulouse point of theindividual Kondo impurities in the original model(3).

B. Relation to the generalized single-impurity model

What are the properties of this fluctuating minidomainand what is the nature of the quantum phase transition sepa-rating the two phases? This question can be tackled withinthe generalized Anderson model(14).

According to Eq.(16) the couplingW in Eq. (14) vanishesfor dJz=dT, i.e., at the Toulouse point. This does, however,not necessarily imply thatW can be treated as a small pa-rameter in the vicinity of the Toulouse point. Rather, it isimportant to realize that two different strong-coupling limitsemerge, depending on the order of limits taken in parameterspace.

(i) If we consider the limitW→0 at fixed smallJ', thenthe critical Kz will diverge according to the analysis above.In this case,Kz is the largest scale, whereasJ' andV can betreated as perturbations.

(ii ) Alternatively, we can takeJ'→0 at fixed smallW,0. Apparently, the criticalKz will vanish (althoughKz/TK

s1d@1), and a strong-coupling treatment has to considerthat W is a large scale.

Both limits correspond to physical situations. However,the scaling limit, discussed at the beginning of Sec. II andemployed in plotting the phase diagram of Fig. 1, is reachedby takingJ'→0 at fixedJz andTKs1d /Kz, corresponding to thecase(ii ) above. More generally, in a three-dimensional phasediagram which involves a third axis labeledJ' in addition tothe ones shown in Fig. 1, case(i) applies for any finiteJ' ina certain vicinity of the Toulouse point, before the behaviorcrosses over to case(ii )—this crossover scale vanishes in theuniversal limitJ'→0.

We will now analyze both cases separately—interestinglythe two limits will turn out to commute.

C. W\0 at fixed V

Assuming thatKz is the largest scale in the problem,which corresponds to a strong local Coulomb repulsion onthe impurity site of the generalized Anderson model, we candirectly map it onto an anisotropic Kondo model. Withn̄Cs= :Cs

†s0dCss0d: and S=1/2da†sabdb, we can rewrite

Wos n̄dsn̄Cs = WSzsn̄C↑− n̄C↓d + sW/2dsn̄d↑+ n̄d↓dsn̄C↑+ n̄C↓d.As charge fluctuations are frozen out for largeKz, the lastterm can be omitted,n̄d↑+ n̄d↓=1, and we obtain

HeffA = H0fCsg + o

nab

JneffSnCa

†s0dsabn Cbs0d, s23d

with Jneff=sJ'

eff ,J'eff ,Jz

effd, where

J'eff =

4V2

Kz, Jz

eff =4V2

Kz+ W. s24d

Spin up and spin down in this Hamiltonian correspond to thetwo states of the minidomain. The phase diagram of Eq.(23)is well known: whenJz

eff is increased from negative values

towards zero(keepingJ'effÞ0 fixed) one observes a quantum

phase transition from a ferromagnetic regime with ln2 re-sidual entropy to a Fermi-liquid phase where the spin isquenched, see Fig. 3. TheS0= ln2 phase can obviously beidentified with our frozen minidomain. It is stable forJz

eff,−uJ'

effu or W,Wc with

Wc = −8V2

Kz. s25d

For Kz→` or V→0, the phase transition is located atWc=0 or equivalentlydJz=dT as anticipated above. Equation(25) is the exact result for the phase boundary forKz→`,provided thatV and W are mapped ontoJ' and Jz as de-scribed in Sec. II. We note again that the limit consideredhere does not correspond to the universal limitJ'→0, be-cause this would giveKz

cr→0 at any fixedW.

D. V\0 at fixed W

Anticipating thatKzcr→0 in this limit, we need to consider

a problem whereW is a large local energy scale. Interest-ingly, at Kz=0 andV=0 the four impurity states are degen-erate even in the presence ofW because of overall particle-hole symmetry. This degeneracy is lifted byKz which (asabove) favors the impurity statesu↑↓l and u↓↑l (assumingKz.0).

We proceed with an analysis similar to the usualSchrieffer-Wolff transformation, now in the presence of alarge W. As usual, hopping processes of second order inVproduce an effective pseudospin interaction between the im-purity and the conduction band. However, as the impurityoccupation in the intermediate states is different from that of

FIG. 3. Schematic renormalization-group flow of the effectiveKondo model(23) (Refs. 1 and 23) describing the screening of theminidomain pseudospin in the limit of largeKz. Here,Jz

eff=0 corre-sponds to the Toulouse point of the original Kondo model. TheFermi-liquid fixed point is characterized by pseudospin screening;the frozen-minidomain phase corresponds to aline of fixed pointswith unscreened pseudospin. The thick line denotes the phaseboundary, given byJz

eff=−uJ'effu. Variation ofJz around the Toulouse

point in the original model(at fixed J' and Kz) corresponds to aparameter variation in the effective model as shown by the dashedline; the dot is the transition point. The transition is in theKosterlitz-Thouless universality class.

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initial and final states, the intermediate state physics involvesan x-ray edge singularity due to the sign change of the strongpotential scattererW.

Analyzing the matrix elements, we find x-ray edge behav-ior which is cutoff byKz, i.e., the effective generated Kondocouplings are of the formV2/Kz

1−b, where

b = − 22

pdW − S 2

pdWD2, s26d

wheredW=pWr /2 is the phase shift fromW.The detailed derivation of the generalized Schrieffer-

Wolff transformation is given in Appendix A. Finally, onearrives at an effective Kondo model of the form(23), butnow with couplings

J'eff =

4V2f'sdWdKz

1−bLb, Jz

eff =4V2fzsdWdKz

1−bLb+ W, s27d

whereL is of the order of the band cutoff, and bothf' andfz are smooth, dimensionless functions ofdW with

limW→0

f'sdWd = f's0d = 1, limW→0

fzsdWd = fzs0d = 1. s28d

As above, the model shows a Kosterlitz-Thouless phase tran-sition in this effective Kondo model occurs at the lineJ'

eff=−uJz

effu, i.e.,

W= −4V2ff'sdWd + fzsdWdg

Kz1−bLb

, s29d

whereb is a function ofW according to 1−b=2s1−ad witha=sJzr−1d2 as introduced above. To obtain an explicit rela-tion betweenW, V, and Kz in the vicinity of the Toulousepoint, we expand inW. Dropping additive logarithmic cor-rections, we have

W1/s1−bd < W, Lb < 1. s30d

With this and Eq.(28) we can rewrite Eq.(29) as

W= − 8V1/s1−adLs1−2ad/s1−ad

Kz, s31d

which is smoothlyconnected to the result(25) obtained inthe limit W→0 of Sec. III C.

As announced, the criticalKz for fixed W depends only onTK

s1d~V1/s1−adLs1−2ad/s1−ad. To fix the prefactor, we first haveto define TK

s1d for an asymmetric Kondo model. This is bestdone using a physical observable such as the impurityspecific-heat coefficientg= limT→0Cimp/T of the anisotropicsingle-impurity Kondo model. We employ

TKs1d ; w

p2

3g−1, s32d

where w=0.4128 is the Wilson number.1 At the Toulousepoint, a=0, one easily findsg=1/s3rFV2d. Thus, forW→0we obtain

rFW= −8

wp2TK

s1d

Kzcr < − 1.964

TKs1d

Kzcr . s33d

For particle-hole symmetric bands, this can be rewritten us-ing Eqs.(15) and (16) into

TKs1d

Kzcr =

rFwp2Î2 sin2S p

2Î2D8

sJzcr − Jzd < 0.578rFsJz

cr − Jzd,

s34d

valid for Jz→Jzcr.Concluding this analysis, we have shown that also in the

limit V→0 (keepingW fixed) the effective Anderson model(14) can be mapped onto an effective Kondo model, whichdescribes the screening of the minidomain pseudospin. Thephase transition is in the Kosterlitz-Thouless universalityclass. The two strong-coupling limits are adiabatically con-nected, as the involved impurity states and transitions aresimilar in both cases; in addition the equations for the phaseboundaries(25) and (31) match.

From the mapping to the anisotropic Kondo model we canalso identify the fluctuating minidomain regime with aFermi-liquid phase. Here, the pseudospin is screened belowthe Kondo temperatureT* of the effective Kondo model(23).Importantly, the Fermi sea of the effective model is formedby solitonic spin excitations of the original model(3)—thiswill strongly influence the conductance through the systemas discussed in Sec. VII. At the Toulouse point, we can esti-mateT* ,minsKz,Ddexpf−uKzu / s8V2rFdg. Close to the quan-tum phase transition, reached forKz/V

2→0 at the Toulousepoint, T* is exponentially small. Similarly, for finiteKz

cr andKz&Kz

cr, one expects for a Kosterlitz-Thouless transition thebehavior34

T* = ae−b/ÎKzcr−Kz, s35d

wherea is a function ofTKs1d andJz; this form will actually be

used to fit the numerical data of Sec. IV.At the Toulouse line, one can investigate the crossover

from the fluctuating minidomain regime to the regime withKondo-screened spins atKz=0 (dashed crossover line in Fig.1). As our model is equivalent to an Anderson impuritymodel (14), this crossover is equivalent to the well-knownAnderson model crossover from mixed-valence to Kondo be-havior. This crossover takes place atKz,V2rF, i.e., whenTK

s1d /Kz is Os1d. Similarly, one finds forJz→` that thiscrossover is located atTK

s1d,J',Kz, and furthermore forJz→0 one also expects this crossover atKz,TKs1d (as noother low-energy scale exists in this regime). We thereforeconclude that the dashed crossover line in Fig. 1 is alwayslocated atTK

s1d /Kz,Os1d.The schematic RG flow of the effective Kondo model,

shown in Fig. 3, illustrates the behavior of the Ising-coupledtwo-impurity model in the strong-coupling regime. Three ob-servations are important.

(i) Both theS0= ln2 frozen-domain phase and theS0=0Fermi-liquid phase are stable to small perturbations—thisconclusion is in agreement with the analysis of Ref. 20.

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(ii ) The S0= ln2 phase corresponds to aline of RG fixedpoints.

(iii ) The two phases are separated by a Kosterlitz-Thouless transition, characterized by logarithmic rather thanpower-law behavior of thermodynamic observables.

So far, the conclusions above mainly apply to the vicinityof the Toulouse point of the single-impurity model, i.e., forstrongly anisotropic Kondo coupling. In the following sec-tion we will present numerical results which strongly supportthat the above picture is valid over the whole phase diagram.In particular, we shall show that the phase transition is ofKosterlitz-Thouless type even in the case of isotropic Kondocouplings. Furthermore, no other phase transition(than theone indicated in Fig. 1) occurs; this implies that the Fermiliquid formed by the fluctuating minidomain is adiabaticallyconnected to the Fermi liquid of two individually Kondo-screened spins, realized in the limitKz!TK

s1d.

IV. NUMERICAL RENORMALIZATION-GROUP ANALYSIS

In this section, we turn to a numerical investigation of themodel of Ising-coupled impurities using the NRGtechnique.2 In principle, an investigation of the original two-impurity model (3) is possible—however, the required twobands of spinful fermions are computationally demandingwithin NRG, and results are significantly less accurate com-pared to the NRG treatment of single-band impurity models.

Therefore, we have decided to study the generalizedAnderson model of Eq.(14), obtained after bosonization andrefermionization of the original model. Featuring only oneband of spinful conduction electrons, it allows high-accuracynumerical simulations down to lowest-energy scales andtemperatures.

A. Parameters

In the following, we will show numerical results for thegeneralized Anderson model(14), for a rectangular particle-hole symmetric fermionic band of widthD=2Î2, and differ-ent values of the hybridizationV, density interactionW, andon-site repulsionKz—note thatKz is identical to the originalIsing interaction between the two impurities, whereasV andW are related toJ' andJz as detailed in Sec. II. In particular,W measures the deviation from the Toulouse point. Accord-ing to Eq. (16), valid for small V, Jz=0 corresponds toWrF=−s2/pdtanfpsÎ2−1d /2g

over temperatureT* below which the pseudospin is screened,see above. For numerical simplicity we definedT* throughSsT*d=0.4. The dependence ofT* on Kz allows us to deter-mine Kz

cr whereT* vanishes. We fitted the data withT*sKzd=aexpfb/ÎKzcr−Kzg (35) with fit parametersa, b, Kzcr, whichis the form expected near a Kosterlitz-Thouless transition.34

The fit works excellently for all anisotropies, as shown inFig. 6—this is again strong support for the Kosterlitz-Thouless nature of the transition.

The single-impurity Kondo temperature,TKs1d for given W

and V is determined from Eq.(32), where the specific-heatcoefficientg is extracted from the NRG data forSsTd.

Having determined bothTKs1d andKz

cr, we are in the posi-tion to plot the phase diagram in the universal fashion indi-

cated in Fig. 1, i.e., employing the limitTKs1d!D. The result

is shown in the main panel of Fig. 7.It is possible to make the meaning of universality more

precise: so far we have distinguished the parameter sets bytheir value of Jz (or W), leading to different values ofTK

s1d /Kzcr. Within the RG treatment of Yuval and Anderson29

FIG. 4. NRG flow diagram for the generalized single-impurityAnderson model(14), for parameter values belonging to(a) theFermi-liquid phase withS0=0, (b) the frozen-domain phase withS0= ln2. Solid—WrF=−0.44, V=0.075sKz

cr=7.6310−10d, dash-dot—WrF=−0.10, V=1.5310

−3sKzcr=3.4310−6d, dashed—WrF

=−0.034, V=1.5310−5sKzcr=4.8310−9d. In (a) and (b), Kz has

been chosen slightly below and above the critical value, respec-tively. For all parameters, the system is in aS=ln4 regime at hightemperatures(small N), in (a) it flows to theS=0 state by passingthrough a regime withS=ln2. In (a), the additional dotted curvesshow the flow forWrF=−0.44,V=0.075, andKz=0. TheWrF val-ues span a large range of anisotropies; nevertheless, theS=0 fixedpoint is unique, and the finite-temperature crossover is universal forthe curves close toKz

cr. Panel(b) nicely shows thatS=ln2 actuallycorresponds to a line of fixed points. NRG parameters areL=2 andNs=650.

FIG. 5. Temperature evolution of the impurity entropy calcu-lated by NRG for the generalized single-impurity Anderson model(14) for different anisotropies of the Kondo coupling. In the frozenminidomain phase the residual entropy is ln2 while it vanishes forKz,Kz

cr. For TKs1d

@Kz (solid curves), the high-temperature ln4 en-tropy is quenched in a single step, whereas two-stage screeningoccurs for TK

s1d,Kz,Kz

cr. (a) WrF=−0.44, V=0.15sKzcr=1.5

310−5d, close to isotropic Kondo coupling.Kz is solid 0, long-dash10−5, long-dash-dot 1.3310−5, short-dash 1.5310−5, short-dash-dot 10−4. (b) WrF=−0.034, V=1.5310

−5sKzcr=4.8310−9d, i.e.,

close to the Toulouse point of the individual Kondo impurities. TheKz values are solid 0, long-dash 10

−9, long-dash-dot 1.5310−9,short-dash 3310−9, short-dash-dot 10−7. (c) WrF=0.44, V=1.5310−7, i.e., on the right-hand side of the phase diagram, Fig. 1,where no phase transition occurs as a function ofKz. Kz is solid 0,long-dash 10−8, long-dash-dot 10−7, short-dash 10−6, short-dash-dot10−5.

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for the single-impurity Kondo model, it is easily seen that wecan expect identical low-energy behavior for two single-impurity models if the initial parameters place the two mod-els on the same RG trajectory.(The RG trajectories are iden-tical to the ones shown in Fig. 3.) Therefore, the correctparameter for the horizontal axis of our phase diagram is aparameter labeling the RG trajectories, i.e., a proper RG in-

variant. Note that forJz.0 andJ'→0, Jz can be used assuch a label—this is what we have done so far. A proper RGinvariant isc defined by29

c = 4sJ'rd2 + e + 2 lnS1 − e2D ,e = 8

dJzp

− 8SdJzpD2, s36d

wheredJz=pJzr /2 is the phase shift resulting fromJz, andwe have employed the bosonization cutoff scheme here. Forsmall values of bothJz andJ', the above equations can beexpanded to yield

c = 4sJ'rd2 − 4sJzrd2. s37d

For J'→0 the value ofc thus depends only onJz as antici-pated; in this regimec,0. The advantage of using a param-etrization of the single-impurity RG flow viac is that it al-lows us to cover the trajectories withuJ' u . uJzu as well, i.e.,the trajectories above the isotropic line in Fig. 3; herec.0.

With the parameter mapping described at the beginning ofthis section we have all parameters at hand and can deter-mine the value ofc from V andW. This allows to replot thephase diagram in a plane spanned byTK

s1d /Kzcr andc—this is

shown in the inset of Fig. 7. In particular, we can now adddata points forJz,0 to the phase boundary plot, as those arecharacterized by afinite J' in the limit TK

s1d!D.As mentioned above, some NRG results show a relatively

strong dependence on the NRG discretization parameterL.Figure 7 shows the phase diagram forL=2; results for otherL values are similar, but theTK

s1d /Kzcr values can differ by

50% or more. Therefore, we have performed an extrapola-tion to L→1 for a few important quantities. A sample ex-trapolation is shown in Fig. 8 for the slope of the phaseboundary near the Toulouse point, which was determinedanalytically in Sec. III—the extrapolated value ofKzW/V

1/s1−ad is consistent with the exact result in Eq.(31).We have also looked at the maximum value ofTK

s1d /Kz of thephase boundary occurring nearJz=0, this value extrapolatesto sTK

s1d /Kzdmax=0.11±0.03.

FIG. 6. Numerically determined values ofT*sKzd for WrF=−0.034, V=1.5310−5 whereKz

cr=4.8310−9 (left), andWrF=−0.44,V=0.075whereKz

cr=7.6310−10 (right); together with theexponential fit described in the text.

FIG. 7. Phase diagram of the generalized single-impurity Ander-son model(14) deduced from NRG calculations for NRG discreti-zation parameterL=2. The vertical dashed line shows the Toulousepoint of the individual Kondo impurities. Small values ofV havebeen used to reach the universal regimeTK

s1d!D. Precise values of

TKs1d have been determined via the specific-heat coefficientg, see

text. The upper horizontal axis shows the corresponding values ofJzin the bosonization cutoff scheme. The error bar shows the typicaluncertainty in the numerical determination ofTK

s1d /Kzcr arising from

the fits of bothg and Kzcr. The inset shows the same data for

TKs1d /Kz

cr, now plotted as a function of the RG invariantc of thesingle-impurity model(36)—this plot covers the range of positiveas well as negativeJz (herec.0). The lines are guide to the eyeonly.

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V. FLOW EQUATIONS

In this section, we consider a different approach39 to ouroriginal model of Ising-coupled Kondo impurities, which isbased on the method of flow equations.35 The general idea isan approximate diagonalization of each of the two Kondoimpurities—the result is a resonant-level-type effectivemodel which captures the Kondo physics in terms of a non-trivial renormalized hybridization. Taking the two Ising-coupled impurities together, we will again arrive at an effec-tive Anderson model. Away from the Toulouse line one findsan additional weak density-density interaction. However, incontrast to Sec. III D where we were forced to treat a similarinteraction nonperturbatively to take into account x-ray edgesingularities(Appendix A), this is not necessary here be-cause this nontrivial physics is already contained in the fullyrenormalized couplings which naturally appear within theflow-equation approach. This approach allows for a system-atic expansion around the Toulouse line, and the effectiveHamiltonian derived in this framework in fact describes theentire phase diagram of Fig. 1 consistently.

A. Flow-equation transformation

The flow-equation method was first applied to the Kondomodel in Ref. 36, where it was shown that it leads to anexpansion around the Toulouse point. Its basic idea is toperform a sequence of infinitesimal unitary transformationson a given many-particle Hamiltonian and thereby diagonal-izing it.35 The expansion parameter turns out to bel0−1 withl0=Î2s1−Jzrd=1−Wr (using the bosonization cutoffscheme); the Toulouse point corresponds tol0=1.

Following Ref. 36, we construct unitary transformationsU1,2 (see Appendix B) such that the single-impurity Hamil-toniansH̃1,2

K from Eq. (10) become diagonal

H1,2sK diagd = U1,2H̃1,2

K U1,2† , s38d

up to higher-order terms in our expansion around the Tou-louse line. For studying the Ising-coupled Kondo impuritieswe therefore apply the combined unitary transformationU

=U1U2 on Eq.(3), H̃K=UHKU†, leading to

H̃K = H1sK diagd + H2

sK diagd + KzS̃1zS̃2

z, s39d

with S̃1,2z =U1,2S1,2

z U1,2† . At the Toulouse point Eq.(39) is of

course exactly equivalent to the Anderson impurity model(13) with W=0 and the same mapping as used in Secs. II Cand II D: the unitary transformationU1,2 just eliminates thehybridization coupling in the Anderson impurity model.

It was shown in Ref. 37 that the flow-equation approachyields a resonant-level model(13) as an effective model forthe Kondo impurity model also away from the Toulousepoint

HsRL effd = H0fC jg + ok

Ṽk„dj†C jskd + H.c.…, s40d

whereC j†skd andC jskd are the creation and annihilation op-

erators for solitonic spin excitations in momentum space.However, this resonant-level model now has a nontrivial

renormalized hybridization function,Dsed =def

ok Ṽk2dse−ekd,with (i) Ds0d=TK

s1d /wp2 and nearly constant in an energyinterval of orderTK

s1d around the Fermi energy(hereeF=0)and (ii ) a nontrivial power-law behavior for larger energies.Furthermore, it was shown in Ref. 37 that toleadingorder inan expansion around the Toulouse line one can identifySj

z

=dj†dj −1/2. The effective model for our system of Ising-

coupled Kondo impurities is therefore an Anderson impuritymodel with a hybridization function of the order of thesingle-impurity Kondo scale:

HsA effd = H0fCsg + ok,s

Ṽk„ds†Csskd + H.c.… + Kzn̄d↑n̄d↓.

s41d

The main feature of the flow equation method is therefore toeliminate the large couplingW in Eq. (14) by renormalizingthe hybridization of the Anderson model.

However, since the flow-equation transformation is an ex-pansion in the distancesl0−1d to the Toulouse line, we needto be careful in the transformation ofSz: The transformedS̃z

is multiplied by a possibly large parameterKz, so that anerror of ordersl0−1d in the expansion becomes multipliedby Kz, leading to additional interaction terms in Eq.(41) thatcan be larger than the hybridization energy scaleTK

s1d.39 It isprecisely these additional interactions that drive theKosterlitz-Thouless transition between the Fermi-liquidphase and the frozen minidomain phase foruKzsl0−1duTK

s1d.

B. Corrections to the transformation of S̃z

In Appendix B the flow-equation solution for the single-impurity Kondo modelH1/2

K in a magnetic fieldh is discussedwith a careful analysis of terms of ordersl0−1d. Hereh is

FIG. 8. L dependence of the slope of the phase boundary nearthe Toulouse point.(L is the NRG parameter defining the logarith-mic discretization of the conduction band.) The dashed line is alinear fit. Each data point involves an extrapolation of the numericalresults at finite negativeW to W→0. A rather strongL dependencecan be observed, however, the extrapolated value appears consistentwith the analytical result(31).

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the effective exchange field due to the second spin, to be

described below. The transformed operatorS̃jz takes the fol-

lowing form:

S̃jz =

1

2E dxdx8dsxdd*sx8dfC j†sxd,C jsx8dg

+1

2sl0 − 1dfshd]xf̄ js0d s42d

plus irrelevant terms(containing, e.g., higher derivatives ofthe bosonic field) and plus higher-order terms of ordersl0−1d2. The first term on the right-hand side of Eq.(42) can beinterpreted as a result of integrating out the hybridizationterm in Eq.(40), while the second term is a correction termnot contained in the original solution in Ref. 36.f̄ js0d de-notes the bosonic spin-density fieldf jsxd without the Fouriercomponents for energies larger thanOsTK

s1dd (with respect tolow-energy properties one does not need to distinguish thesefields). Properties of the dimensionless functionfshd are de-rived in Appendix B, in particular,fshd=vF / uhu+Osh−2d foruhu@TK

s1d.In the coupled system(41) we can approximate the effect

of one spin on the other as a static magnetic field of strengthh= ±Kz/2 close to the transition. This approximation be-comes asymptotically exact as one approaches the transitionsince the spin dynamics becomes slower and slower.

We arrive at the following Hamiltonian describing thecoupled Kondo impurities in the vicinity of the transitionline:

HsA effd < H0fCsg + ok,s

Ṽk„ds†Csskd + H.c.… + Kzn̄d↑n̄d↓

+ sl0 − 1dKz2

fsKz/2df]xf̄↑s0dn̄d↓ + n̄d↑]xf̄↓s0dg,

s43d

up to corrections of ordersl0−1d2.

C. The Kosterlitz-Thouless transition

Let us now focus on the caseKz@TKs1d that is relevant for

studying the phase transition in the vicinity of the Toulouseline. Using fshd

VI. SYMMETRIES AND PERTURBATIONS

To what extent do the results presented in the previoussections depend on the details of the models under consider-ation? To answer this question we will investigate whetherand how(small) perturbations of Eq.(3) will qualitativelychange the physics. Fermi-liquid phases with vanishing re-sidual entropyS0 are stable against small perturbations. Thisis not necessarily the case for our frozen minidomain char-acterized byS0= ln2. The existence of thisS0= ln2 phase is afundamental feature of our model(3). The necessary condi-tions for its stability will be discussed in what follows.

First, let us consider the effect of a magnetic field inzdirection acting on the impurity spins. A staggered magneticfield, hssS1

z−S2zd, will directly destroy the degeneracy of the

two configurationsu↑↓l, u↓↑l. However, in the limitKz→` ahomogeneous magnetic fieldhsS1

z+S2zd will not destroy the

S0= ln2 phase. It is interesting how these terms modify thegeneralized Anderson model(14). The magnetic fieldh re-sults in a termhos ds

†ds, which breaks particle-hole symme-try in the generalized Anderson model. It therefore modifiesonly the position of the phase boundary. The staggered mag-netic field hs, however, will lead to a termhsos sds

†ds,which corresponds to a(pseudo) magnetic field acting on thepseudospin of the Anderson model. Only the staggered mag-netic field is a relevant perturbation which destroys the ln2phase.

Apart from these magnetic fields inz direction there areother relevant terms which lift the twofold degeneracy,which are of the forms

Sj+, j = 1,2, s52d

S1+S2

−, s53d

S1+S2

−Cis† C js, i, j = 1,2, s54d

S1+S2

−Cia† sabC jb, i, j = 1,2, s55d

and their Hermitian conjugates. It turns out that all theseoperators are forbidden if we impose the following two sym-metry conditions: The model should be invariant under thetwo separate spin rotations of each impurity and its elec-tronic bath about an angle ofp, i.e., under the transformation

Uj = eipI j

z, s56d

with j =1,2. I jz is the z component of spin of systemj , I j

z

=Sjz+ok cka j

† 1/2sabz ckb j. In the presence of thesep rotation

symmetries,Uj, the terms(52)–(55) are absent and the fro-zen minidomain phase survives. The quantum phase transi-tion from the frozen minidomain with residual entropy ln2 tothe phase of Kondo screened impurities therefore just relies(in the absence of a staggered magnetic field) on the symme-tries U1 andU2.

The model(3) considered in this paper possesses by con-struction symmetries beyondUj. They are not necessary forthe stability of theS0= ln2 phase. For example, the two bathsare assumed to have the same Kondo couplingJn. This paritysymmetry can be relaxed without destroying the frozen mini-domain phase. Furthermore, thez component of spin of eachsystem,I j

z, is conserved in our model since we choseJx=Jy=J'. This symmetry can also be perturbed without lifting thetwofold degeneracy. Moreover, the frozen minidomain phaseis stable against breaking of particle-hole symmetry whichwe implicitly assumed in the bosonization treatment by lin-earizing the dispersion relation of the conduction electrons.In all these situations, we therefore expect that all of thequalitative results, i.e., the structure of the phase diagramand the nature of the quantum phase transition, are not af-fected.

However, any perturbation which breaks eitherU1 or U2(or both) will generically generate one of the relevant cou-plings (52)–(55) which all destroy the ln2 phase. In the fol-lowing we briefly discuss two such cases which are likely tooccur in experimental realizations[a third case, correspond-ing to Eq.(54) is studied in Sec. VII A].

First, consider a situation where a small spin-flip coupling(53) is added on top of the large Ising interaction of thespins,

dH12' = K'sS1

xS2x + S1

yS2yd. s57d

In realizations of our model based on spins and stronglyanisotropic spin-orbit interactions, such a term will alwaysbe present. A smallK' will immediately lead to a tunnelingbetween the two states of our minidomain: their degeneracyis lifted, the two spins form a singlet and the ln2 residualentropy is quenched completely. Two-impurity Kondo mod-els with K'=Kz have been widely studied.

12–19As argued inRefs. 16 and 17 the resulting phase diagram depends on thepresence or absence of particle-hole symmetry(which does,however, not modify the phase diagram forK'=0 as pointedout above). In the absence of particle-hole symmetry, thephase transition is replaced by a smooth crossover. However,in the presence of particle-hole symmetry, the scatteringphase shifts of the electrons can only take the values 0 orp /2. As the Kondo-screened phase and the interimpurity sin-

FIG. 9. Phase diagram of the generalized single-impurity Ander-son model(14) deduced using the flow equation method. Notationis as in Fig. 7.

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glet phase have different phase shifts, there has to be a phasetransition in between. This transition is not of Kosterlitz-Thouless type, but characterized16,18 by a residual entropylnÎ2. Nevertheless, this transition will merge with ours in thelimit K'→0, as an infinitesimalK' does not affect theKondo-screened phase but leads immediately to the forma-tion of an interimpurity singlet in the frozen minidomainphase.

A second interesting case is a situation where the twoFermi seas are coupled, e.g., by a tunneling between the twoleads

dH12tunneling= o

k,k8,a

stkk8:cka1† ck8a2: + H.c.d. s58d

While this term is not relevant by power counting, it willinduce a RKKY interaction between the spins and thereforegenerate the relevant coupling(53) and(57). As such a termalso breaks particle-hole symmetry, the quantum phase tran-sition will be replaced by a smooth crossover.

VII. TRANSPORT

In this section we illustrate how the phase diagram and,more importantly, the corresponding quantum phase transi-tion can be revealed in transport experiments. We shall dis-cuss two experimental setups.

(a) Capacitively coupled quantum dots, where thecharge degrees of freedom play the role of(pseudo)spins,8,9

are a promising realization of our model. By adding a smallinter dot tunneling term, we obtain a characteristic zero-biasanomaly.

(b) If the Ising coupling is realized between real spindegrees of freedom, then we shall show that a transport ex-periment can reveal a universal fractional critical conduc-tance at the phase transition which is related to the universaljump of the superfluid density at the Kosterlitz-Thoulesstransition of superfluid thin films. Using quantum dots thissituation is difficult to achieve, as a transverse spin couplingwill always be present in the experiment. Nevertheless, thefollowing proposals highlight the nontrivial effects of aKosterlitz-Thouless(KT) transition with a bath of solitonicparticles onto the original electrons.

A. Zero-bias anomaly of capacitively coupled quantum dots

In realization of our model(3) usingchargestates of ca-pacitively coupled quantum dots8 the conductance discussedabove cannot be easily measured. We therefore propose an-other experiment sketched in Fig. 10. We consider two largequantum dots, each coupled to a(single-channel) lead. TheCoulomb interaction and an appropriately chosen gate volt-age ensure that two charge states on each dot are degenerate,and that all other charge states have higher energies. Thesetwo charge states in each dot take over the role of the twospins as explained in more detail by Matveev.8 The spin-upand- down states of the conduction electrons in our modelcorrespond to electrons sitting either in the leads or on thedot, where we assume that the level spacing on this dot issmall compared to temperature. Using this mapping, a ca-

pacitive coupling of the two dots directly corresponds to anIsing coupling(1). The physical spin in such a system wouldtranslate to an extra channel index in our model. For simplic-ity we will, however, consider a situation where either strongspin-orbit scattering mixes those channels or where the spinis quenched by a strong magnetic field—in both cases weeffectively deal with spinless fermions and thus with asingle-channel model.

We now consider a situation where the two dots arecoupled by weak tunnelingt in addition to the large inter dotcapacity. This tunneling takes the form

Htun = tS1+S2

−C↓1† s0dC↓2s0d + H.c. s59d

We assume that the tunneling is sufficiently weak such that itcan be treated perturbatively in the experimentally relevant

FIG. 10. (a) Experimental setup to measure the tunneling con-ductance between two capacitively coupled quantum dots.(b) Sche-matic plot of the zero-bias anomaly of the conductance atT=0. Inthe frozen minidomain phase,d,dc, the conductance diverges al-gebraically according to Eq.(61). At the quantum phase transition,d=dc, the exponent takes the universal value −2sÎ2−1d accordingto Eq.(62). In the Kondo-screened phase,d.dc, the conductance isfinite for V→0.

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temperature range. This is precisely the situation which wasalso considered by Andreiet al. 20 Note that the approxima-tion to consider only tunneling intoCsisx=0d in Eq. (59) isonly valid if the tunnel contact between the dots is suffi-ciently close to the lead contact[see Fig. 10].

We calculate the conductance in perturbation theory in theinterdot tunnelingt starting from the Kubo formula. The cur-rent through the link between the dots is then given byj= tS1

+S2−fiC↓1

† s0dC↓2s0dg+H.c. and theT dependence of thecurrent-current correlator can be obtained from simple powercounting.

We first consider the frozen minidomain phase. Followingthe arguments given in Sec. III, the dimension of the tunnel-ing term (or equivalently of the current operator) with re-spect to this fixed point is given by dimfHtung=dimf jg=1−s2d /pd2−s1−2d /pd2. Therefore the current-current cor-relator decays in time ast−2sdimf jg−1d, and we obtain for theconductance

GsTd , t2T−2dimf jg = t2T−4s2d/pds1−2d/pd. s60d

This divergence of the conductance arises because the tun-neling is arelevant perturbation which will finally destroythe frozen minidomain phase and quench its residual entropyln2 below some small energy scale. Equation(60) is there-fore only valid for sufficiently smallt when this scale issmaller thanT. Furthermore, a finite domain-flip rate inducedby Eq. (18) is required to obtain a finite current. Above weimplicitly assumed thatt is so small that the size of thecurrent is solely determined by the smallest bottleneck forcharge transport given byt.

At finite voltageV@T, T in Eq. (60) can be replaced byVand we expect a zero-bias anomaly characterized by a pro-nounced peak in the conductance:

GsVd , uVu−4s2d/pds1−2d/pd. s61d

Upon approaching the quantum phase transition, the diver-gence increases and at the KT transition takes the universalform

GcrsTd , T−2sÎ2−1d < T−0.83, s62d

GcrsVd , uVu−2sÎ2−1d < uVu−0.83, s63d

up to logarithmic corrections.In the Kondo-screened phase, we can calculate the quali-

tative behavior of the current-current correlator at the pointin the phase diagram whereKz=0 and the dots decouple. Thecurrent-current correlator then can be decomposed into twocorrelators of the formkSi

+stdC↓i† stdSi

−s0dC↓is0dl, which de-cay asymptotically as 1/t. This can be seen if one identifiesthis correlator with the conduction electronT matrix (seeRef. 40, and references therein) which is characterized by aconstantspectral density for low energies. The conductancetherefore approaches a constant for temperatures and volt-ages well below the characteristic crossover temperatureT*

to the Kondo-screened phase:

GsVd < GsTd < const. s64d

In Fig. 10 we show schematically the nonlinear conductanceas a function ofV in the vicinity of the quantum phase tran-sition.

In contrast to Eqs.(60) and(64), Andrei et al. 20 obtainedan exponentially small conductance in the frozen minido-main phase andG,T4 in the singlet phase, with which wedisagree.

B. Universal conductance of Ising-coupled quantum dots

What is the most characteristic signature of the Kosterlitz-Thouless quantum phase transition which we found in theprevious sections? The most famous example of a Kosterlitz-Thouless transition is probably the vortex binding-unbindingtransition in superfluid4He films. From the Kosterlitz-Thouless theory follows the prediction of a universal jump inthe superfluid density upon passing through the transition.38

Interestingly, the analog of the superfluid density in ourmodel is the scattering phase shiftd of the conduction elec-trons, and the arguments for a universal jump in the super-fluid density carry over to a universal jump ind. This can beseen by considering the RG flow diagram in Fig. 3: In thefrozen minidomain phase the system flows towards a line offixed points which is naturally characterized by the dimen-sion of the leading irrelevant operator, i.e., the domain flip(18), or, equivalently, according to Eq.(21) by the phaseshift d of the conduction electrons. Upon approaching thequantum phase transition, the irrelevant domain flips becomemarginal and the phase shift increases and reachesdT=p /2s1−1/Î2d at the phase boundary[see Eq.(22)]. On theother side of the phase diagram, the system flows to thestrong-coupling fixedpoint, where the Kondo spins arescreened and the electrons acquire a phase shift ofp /2.Therefore the phase shift jumps across the transition fromdTto p /2! This picture is expected to hold everywhere close tothe phase boundary as long as no other phase transitionintervenes—that the latter does not happen is shown by ourNRG calculations.

This universal jump of the phase shift has direct experi-mental consequences. Consider the experimental setupsketched in Fig. 11 where the conductance through the leftdot is measured. If Kondo screening prevails, the conduc-tance forT→0 will be given by the conductance quantumG0=2e

2/ s2p"d. In the frozen minidomain phase on the otherside of the phase diagram, spin flips are completely sup-pressed forT→0 and therefore we can assume astatic spinconfiguration to calculateGsT=0d. For such a potential scat-tering problem, the conductance is given by

GsT = 0d = G0 sin2d. s65d

Directly at the quantum phase transition, the conductancetherefore takes the universal value

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GcrsT = 0d = G0 sin2dT = G0 cos2F p2Î2G < 0.2G0,

s66d

and it jumps to the Kondo valueG0 upon entering theKondo-screened phase. This universal fractional conductanceat our quantum phase transition is one of the remarkableresults of this paper.

It is interesting to compare this to the well-known resultfor the usual Kondo effect, where the conductance jumpsfrom 0 to G0 when the exchange couplingJ is tuned fromferromagnetic to antiferromagnetic. Both in Secs. III D andV C we mapped our model close to the quantum phase tran-sition to such a Kondo model. The fermionic degrees of free-

dom in these Kondo models[Eqs. (23) and (45)] are, how-ever, complexsolitonic excitations in terms of the originalfermions. While the phase shifts of those solitons vanishes atthe quantum phase transition, the phase shift of thephysicalelectrons takes the fractional valuedT leading to a fractionalconductance. Also in other systems which are described by aKosterlitz-Thouless quantum transition in terms of solitons, auniversal fractional conductance of similar origin can be ex-pected at the transition.

In Fig. 11 the zero-temperature conductance close to thephase transition is shown. At any finite temperatures, thejump in the conductance is strongly smeared as sketchedschematically in the figure. TheT dependence at lowest tem-perature is determined by the dimension of the leading irrel-evant operators. In the Kondo-screened phase, the leadingcorrections forT→0 to the Kondo conductanceG0 are oforder sT/T*d2 for T!T* , where T* is exponentially smallclose to the quantum phase transition(see Fig. 6). In thefrozen minidomain phase, corrections to Eq.(65) vanish asT−2dimfHeff

flipg, where dimfHeffflipg is defined in Eq.(21). Directly

at the quantum phase transition the exponent vanishes andleading corrections to Eq.(66) are of the order 1/ lnT andtherefore rather large.

VIII. SUMMARY

We have investigated a model of two Ising-coupledKondo impurities using strong-coupling expansion, numeri-cal renormalization-group calculations, and a transformationbased on the method of flow equations. Those methods yieldconsistent results and allowed us to show the existence of aKosterlitz-Thouless phase transition between a Fermi-liquidphase and a pseudospin doublet phase which corresponds toa frozen minidomain. This transition can be tuned both byvarying the Ising coupling between the impurities and byvarying the anisotropy of the individual Kondo couplings. Inparticular, at the Toulouse point of the individual Kondo im-purities we could map the modelexactly to an Andersonimpurity model with a Fermi sea consisting of fermionicsoliton excitations—in this situation no phase transition oc-curs, and the system is in the Fermi-liquid phase, where theimpurity pseudospin is screened below a collective KondoscaleT* . For Jz smaller than the Toulouse point value, largeKz drives the system into the pseudospin doublet phase.

The most promising way to realize our model is the situ-ation of capacitively coupled quantum dots where the impu-rity spins represent charge degrees of freedom on the dots.We have shown that a small additional tunneling between thedots gives rise to a zero-bias conductance anomaly with auniversal fractional power-law occurring at the transitionpoint. In addition, we have discussed a setup which is inter-esting on theoretical grounds, namely transport through onequantum dot of a pair of dots with a magnetic Ising coupling,where we have found a universal fractional conductancethrough the device at the phase transition point.

With an eye towards comparison with experiments wediscuss the finite-temperature crossover behavior across thephase diagram(see also Fig. 5). If we fix the parameters ofthe individual Kondo impurities, then varyingKz corre-

FIG. 11. (a) Experimental setup to measure the conductancethrough a single quantum dot, which is Ising coupled to a seconddot. The couplings to the leads and between the dots can be tunedusing appropriate gate voltages.(b) At T=0, the conductance(solidline) takes at the quantum phase transition the universal valueGcr=G0 cos

2p /2Î2, Eq. (66). Dashed line: schematic plot of the con-ductance at finiteT. Corrections to theT=0 result are logarithmic atthe transition. The exponent 2d;−2dimfHeff

flipg is given by the di-mension of the domain-flip term(18).

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sponds to a vertical cut through the phase diagram in Fig. 1;the resulting finite-temperature behavior is sketched in Fig.12.

For small Kz there is a single crossover at the single-impurity Kondo temperatureTK

s1d. This crossover splits intotwo whenKz approaches values of orderTK

s1d—then the de-scribed two-stage quenching of the entropy is observed. Theupper crossover temperatureT0 is associated with the forma-tion of the magnetic minidomain, where relative fluctuationsof the two impurity spins are frozen out. The lower crossovertemperature is the collective Kondo scaleT* below which thepseudospin of the minidomain is screened.T* becomes ex-ponentially small nearKz

cr and vanishes forKzùKzcr. For

KzùKzcr another crossover line appears which, however, has

much weaker signatures, namely, the character of the leadingcorrections to the entropy and other quantities changes, as iseasily understood from the RG flow in Fig. 3. For largeKzthe entropy change from ln4 to ln2 occurs aroundT,Kz,thereforeT0 approachesKz in this limit.

Interestingly, the different impurity degrees of freedomcan be reinterpreted: the flipping of the pseudospin whilekeeping the minidomain intact apparently corresponds topseudospin “phase” fluctuations, whereas breaking up theminidomain is related to “amplitude” fluctuations of thepseudospin. Thus, in Fig. 12 we encounter the situation thatamplitude fluctuations are frozen out at a higher temperatureT0 whereas phase fluctuations are quenched at the lowerT

* ,in other words, the two impurity spins fluctuate indepen-dently forT.T0 whereas they fluctuate in a correlated fash-ion betweenT* ,T,T0. This physics is surprisingly similarto the behavior of lattice systems in low dimensions, with the

difference that of course no true ordering can occur in theimpurity model.

In summary, the present two-impurity model shows re-markably rich behavior, which awaits realizations in mesos-copic devices. An interesting extension would be the two-channel case which is naturally met in capacitively coupleddots with spin-degenerate conduction electrons.

ACKNOWLEDGMENTS

The authors acknowledge fruitful discussions with N. An-drei, R. Bulla, L. I. Glazman, K. Le Hur, and A. Schiller.This work was supported by the Deutsche Forschungsge-meinschaft(DFG) through SFB 484(S.K.,T.P.), the Emmy-Noether program(M.G.,A.R.), and the Center for FunctionalNanostructures at the University of Karlsruhe(M.V.). S.K.also acknowledges support by the DFG.

APPENDIX A: GENERALIZED SCHRIEFFER-WOLFFTRANSFORMATION

In this appendix, we perform explicitly the mapping of thegeneralized Anderson model(14) to the Kondo Hamiltonian(23) for largeKz. Due to the presence of the interactionW inEq. (14) the usual Schrieffer-Wolff transformation has to begeneralized to take into account power-law renormalizations(of “x-ray edge” type) induced byW. We derive the mappingby investigating directly the properties of a perturbative ex-pansion in the hybridizationV for finite W within the Ander-son model. Consider the generalized Anderson model in itsbosonized version. We first eliminate theW term in Eq.(14)by the Emery-Kivelson transformation(9) with g* =Wr andobtain

Ug*HAUg*

† = os

H0ffsg + Kzn̄d↑n̄d↓ + Hint, sA1d

whereW enters only the hybridization term

Hint =V

Î2paos sds†e−is1−Wrdfss0dFs + H.c.d. sA2d

For largeKz the d level is only singly occupied.V inducesvirtual fluctuations to the doubly occupied and empty statewhich are separated from the singly occupied statesu↑ l, u↓ lby an energyKz/2. To derive the effective Kondo modelconsider theS matrix with respect to this low-energy sub-space:

T expf− iE−`

`

dtHintstdg

=on=0

` E−`

`

dt2n ¯ dt1t2n.¯.t1

iH intst2nd ¯ iH intst1d, sA3d

Hint describes processes from the low-energy sector to highenergies or back. Such virtual excitations are rare and existonly for a short time ifKz is large. Therefore we can groupthem to pairs to obtain an effective interaction living in thelow-energy Hilbert space,

FIG. 12. Schematic phase diagram as a function ofKz andT forfixed TK

s1d. For T=0 there is a quantum phase transition atKz=Kzcr

from a Fermi liquid with residual entropyS0=0 to the frozen mini-domain phase withS0= ln2. At T.0, only smooth crossovers occurindicated by the dashed and dotted lines. At the dashed lines, theentropyS changes by ln2(see also Fig. 5), while at the dotted lineone obtains a crossover from a logarithmic to a power-law behaviorin the leading corrections toS. Similar crossovers also occur intransport quantities. BelowT0 a magnetic minidomain is formed,while a Fermi liquid is recovered belowT* which is exponentiallysmall close toKz

cr.

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E−`

t2m+1dt2mE

−`

t2mdt2m−1iH intst2mdiH intst2m−1d

< − iE−`

Tm+1Hint

effsTmd, sA4d

with

HinteffsTmd = − iE

0

`

dtHintsTm + t/2dHintsTm − t/2d,

where we introduced the center of time and relative coordi-nates. Interactions between adjacent virtual excitations canbe neglected to leading order for largeKz. Introducing thespin notation,Sz=1/2os sds

†ds and S+=d↑

†d↓, to representthe two states of the low-energy Hilbert space, the aboveexpression becomes

HinteffsTmd = − i

V2

2paE

0

`

dte−iKzt/2os

fe−i2Szss1−WrdfssTm+t/2d

3ei2Szss1−WrdfssTm−t/2d + sS+F↑F↓†e−iss1−WrdfssTm+t/2d

3eiss1−Wrdf−ssTm−t/2d + H.c.dg.

The oscillating factore−iKzt/2 guarantees that the virtual exci-tations are only short lived, and we can therefore expand theterm in the bracket in the small timet. Introducing the spinfield f=s1/Î2dos sfs, applying the operator product ex-pansion eilfsstde−ilfsst8d=f1+ist− t8d /ag−l

2+laf1+ist

− t8d /ag1−l2]t8fsst8d+¯ for the first term, integrating overt

using e0` dte−iKzt/2s1+it /ad−a=−isaKz/2da2Gs1−ad /Kz we

obtain in leading order for largeKz,

Hinteff =

4V2

KzÎ2ps1 − WrdGf2 − s1 − Wrd2g

3SaKz2Ds1 − Wrd2−1Sz:]xfs0d:

+4V2

Kz2pasS+e−iÎ2s1−Wrdfs0dF↓

†F↑ + H.c.d. sA5d

Before identifying the coupling constants of the effectivelow-energy Hamiltonian two more steps are required. First,we have to readjust our UV cutoff froma to aK,1/Kz in thedefinition of our fields, as we effectively have integrated outshort-time differences of order 1/Kz. To this end we have tonormal-orderHint

eff as only normal-ordered expressions arecutoff independent,

eilf = S2paL

Dl2/2:eilf:=S aaKDl2/2S2paK

LDl2/2:eilf:

= S aaKDl2/2eilf̃, sA6d

wheref̃ denotes the fields defined with respect to the newcutoff aK. This effectively leads to the substitution

4V2

Kz2pa→ 4V

2

Kz2paKS a

aKDs1 − Wrd2−1 sA7d

in the second term of Eq.(A5).In a last step, we undo the Emery-Kivelson transforma-

tion to obtain the Kondo Hamiltonian in its usual form(23)with

Jzeff = W+

4V2

KzcWSaKz2 D

s1 − Wrd2−1

, sA8d

J'eff =

4V2

KzS a

aKDs1 − Wrd2−1, sA9d

wherecW=s1−WrdGf2−s1−Wrd2g. X-ray edge singularitiesinduced byW have led to a power-law dependence of theeffective couplings onKz. Note that the previous argumentsfixed aK,1/Kz in Eq. (A9) only up to a prefactor of order 1depending onW. However, this unknown prefactor ap-proaches 1 close to the quantum phase transition whereWr→0.

APPENDIX B: FLOW EQUATION TRANSFORMATIONFOR THE SINGLE KONDO IMPURITY

In this appendix we provide some details on the flow-equation treatment of the single-impurity Kondo model,which was first presented in Ref. 36. Here we will show howto extend the analysis of Ref. 36 to take into account theterms of ordersl2−1d that become important in our coupledsystem since they are multiplied by a possibly large energyscaleKz in Eq. (41). We refer the reader to Ref. 36 for thebasic ideas of the approach and only present the main stepsto keep our presentation here self-contained.

1. Transformation of the Hamiltonian

The starting point for the flow-equation approach is Eq.(10) with g chosen such that the longitudinal coupling iseliminated. This way we arrive at the initial HamiltonianHsB=0d for the flow-equation approach,

HsBd = H0ffg +E dxgsB;xdfV„lsBd;x…S− + H.c.g,sB1d

with lsB=0d=l0=Î2−Jz/Î2pvF and

gsB = 0;xd = dsxdS2paL

DlsB = 0d2/2 J'2pa

. sB2d

HereVsl ;xd are normal-ordered vertex operators

Vsl;xd = :e−ilfsxd:. sB3d

During the course of the infinitesimal unitary transformations

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dHsBddB

= fhsBd,HsBdg, sB4d

with the generatorhsBd from Ref. 36,

h =E dxhs1dsxdfVsl;xdS− − H.c.g+E dxdx8hs2dsx,x8dfVsl;xd,Vs− l;x8dg, sB5d

the interactiongsB;xd in Eq. (B1) becomes more and morenonlocal. With each infinitesimal step of the transformationone also generates a new interaction term in Eq.(B1) withthe structure

vFSzE dxssxd]xfsxd sB6d

and a nonlocal functionssxd that depends on the couplings.The key step in Ref. 36. is that Eq.(B6) can again be elimi-nated by a unitary transformation of the Emery-Kivelsontype,

U = :expSiSzE dxssxdfsxdD:. sB7dWe now analyze how the interaction term in Eq.(B1) is

transformed due toU, e.g.,

UVsl;ydS−U† = :expS− i2E dxssxdfsxdD

3::e−ilfsyd::expS− i2E dxssxdfsxdD:S−.

sB8d

In order to proceed we normal order all the exponentials,which can be done exactly since the commutator of thebosonic field is ac number. This leads to

UVsl;ydS−U† ~ :expSÎ2pL ok.0 e−ka/2

Îkhfle−iky + sskdgbk

− fleiky + sskdgbk†jD:S−, sB9d

with sskd being the Fourier transform ofssxd from Eq. (B6).The proportionality factor in(B9) leads to the nonperturba-tive renormalization of the coupling constantgsB;xd alreadyobtained in Ref. 36. Except for the local coupling at thebeginning of the flow the exponential in Eq.(B9) cannot beexactly rewritten as a vertex operator. We use two approxi-mations that give us the correct result up to quadratic termsin the deviation from the Toulouse line

(i) We use the infrared limitss0d instead ofsskd in Eq.(B9).

(ii ) We expand the exponential in a way that avoids IRdivergences and neglect higher-order terms in the bosonicoperators that lead to irrelevant couplings:

:expSÎ2pL ok.0 e−ka/2

Îkhfle−iky + ss0dgbk − H.c.jD :

=:expSÎ2pL ok.0 e−ka/2

Îk„fhl + ss0dje−iky

+ s1 − e−ikydss0dgbk − H.c.…D :=:V„l + ss0d;y…S1 +Î2pL ok.0 e

−ka/2

Îk

3fs1 − e−ikydss0dbk − H.c.g + ¯D:. sB10dRetaining only the first term on the right-hand side is theapproximation used in Ref. 36: one obtains vertex operatorswith flowing scaling dimensionsfl+ss0dg that eventuallybecome fermionic. The second term can be understood as acorrection term to this leading behavior